Geography Reference
In-Depth Information
7.6. In Section 7.3.1 the linearized equations for acoustic waves were devel-
oped for the special situation of one-dimensional propagation in a horizontal
tube. Although this situation does not appear to be directly applicable to the
atmosphere, there is a special atmospheric mode, the Lamb wave , which is
a horizontally propagating acoustic mode with no vertical velocity pertur-
bation (w
0). Such oscillations have been observed following violent
explosions such as volcanic eruptions and atmospheric nuclear tests. Using
(7.12), (7.13) plus the linearized forms of the hydrostatic equation, and the
continuity equation (7.7) derive the height dependence of the perturbation
fields for the Lamb mode in an isothermal basic state atmosphere, assuming
that the pressure perturbation at the lower boundary (z
=
0) has the form
(7.15). Determine the vertically integrated kinetic energy density per unit
horizontal area for this mode.
=
7.7. If the surface height perturbation in a shallow water gravity wave is given by
Re Ae ik(x ct)
h =
find the corresponding velocity perturbation u (x, t). Sketch the phase rela-
tionship between h and u for an eastward propagating wave.
7.8. Assuming that the vertical velocity perturbation for a two-dimensional inter-
nal gravity wave is given by (7.43), obtain the corresponding solution for
the u , p , and θ fields. Use these results to verify the approximation
p /c s
ρ 0 θ
which was used in (7.36).
7.9. For the situati on in Problem 7.8, express the vertical flux of horizontal
momentum, ρ 0 u w , in terms of the amplitude A of the vertical velocity
perturbation. Hence, show that the momentum flux is positive for waves in
which phase speed propagates eastward and downward.
7.10. Show that if (7.38) is replaced by the hydrostatic equation (i.e., the terms in
w are neglected) the resulting frequency equation for internal gravity waves
is just the asymptotic limit of (7.44) for waves in which
| k | | m |
.
7.11. (a) Show that the intrinsic group velocity vector in two-dimensional internal
gravity waves is parallel to lines of constant phase. (b) Show that in the
long-wave limit (
) the magnitude of the zonal component of the
group velocity equals the magnitude of the zonal phase speed so that energy
propagates one wavelength per wave period.
|
k
| |
m
|
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